Mathematical modelling of the spread of HIV/AIDS amongst injecting drug users

In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. Our work is based on a model originally due to Kaplan (1989, Rev. Inf. Diseases11, 289–98). We start off with a brief literature survey and review; this is followed up by a detailed description of Kaplan's model. We then outline a more realistic extension of Kaplan's model. Then we perform an equilibrium and stability analysis on this model. We find that there is a critical threshold parameter R0 which determines the behaviour of the model. If R0 ≤ 1 there is a unique disease-free equilibrium, and if R0 < 1 the disease dies out. If R0 > 1 this disease-free equilibrium is unstable, and in addition there is a unique endemic equilibrium which is locally stable. If a certain condition is satisfied (and for Kaplan's model this condition is always satisfied), additional complete global-stability results are shown. These results are confirmed and explored further by simulation.